On automatic tuning of basis functions in Bezier method

Journal of Computational and Applied Mathematics, 2008

A new formulation for the representation and designing of curves and surfaces is presented. It is a novel generalization of Bézier curves and surfaces. Firstly, a class of polynomial basis functions with n adjustable shape parameters is present. It is a natural extension to classical Bernstein basis functions. The corresponding Bézier curves and surfaces, the so-called Quasi-Bézier (i.e., Q-Bézier, for short) curves and surfaces, are also constructed and their properties studied. It has been shown that the main advantage compared to the ordinary Bézier curves and surfaces is that after inputting a set of control points and values of newly introduced n shape parameters, the desired curve or surface can be flexibly chosen from a set of curves or surfaces which differ either locally or globally by suitably modifying the values of the shape parameters, when the control polygon is maintained. The Q-Bézier curve and surface inherit the most properties of Bézier curve and surface and can be more approximated to the control polygon. It is visible that the properties of end-points on Q-Bézier curve and surface can be locally controlled by these shape parameters. Some examples are given by figures.

Computer Aided Geometric Design, 2002

It is proved that the C-B-spline segments introduced by Zhang are generated by a normalized totally positive basis. It is also constructed a normalized B-basis of C-B-splines, which presents optimal shape preserving and stability properties.

2004

The Bezier curve is fundamental to a wide range of challenging and practical applications such as, computer aided geometric design, postscript font representations, generic object shape descriptions and surface representation. However, a drawback of the Bezier curve is that it only considers the global information of its control points; consequently, there is often a large gap between the curve and its control polygon, which leads to a considerable error in curve representations.

International Journal for Numerical Methods in Engineering, 2011

We develop finite element data structures for T-splines based on Bézier extraction generalizing our previous work for NURBS. As in traditional finite element analysis, the extracted Bézier elements are defined in terms of a fixed set of polynomial basis functions, the so-called Bernstein basis. The Bézier elements may be processed in the same way as in a standard finite element computer program, utilizing exactly the same data processing arrays. In fact, only the shape function subroutine needs to be modified while all other aspects of a finite element program remain the same. A byproduct of the extraction process is the element extraction operator. This operator localizes the topological and global smoothness information to the element level, and represents a canonical treatment of T-junctions, referred to as 'hanging nodes' in finite element analysis and a fundamental feature of T-splines. A detailed example is presented to illustrate the ideas. Figure 2. A schematic diagram illustrating the central role played by Bézier extraction in unifying CAGD and FEA.

2001

Springer eBooks, 2022

The univariate minimal support B-spline basis (UMB) has been used in Computer Aided Design (CAD) since the 1970s. Freeform curves use UMB, while sculptured surfaces are represented using a tensor product of two UMBs. The coefficients of a B-spline curve and surface are respectively represented in a vector and a rectangular grid. In CAD-intersection algorithms for UMB represented objects, a divide-and-conquer strategy is often used. Refinement by knot insertion is used to split the objects intersected into objects of the same type with a smaller geometric extent. In many cases the intersection of the resulting sub-objects has simpler topology than the original problem. The sub-objects created are represented using their parents' UMB format and deleted when the sub-problem is solved. Consequently, no global representations of the locally refined bases are needed. This is contrary to when locally refined splines are used for approximation of large point sets. As soon as a B-spline is locally refined, the regular structure of UMB objects in CAD is no longer valid. In this chapter we discuss how Locally Refined B-splines (LR B-splines) address this challenge and present the properties of LR B-splines. Keywords Locally Refined B-splines • Minimal support basis • Refinement

2009 Sixth International Conference on Computer Graphics, Imaging and Visualization, 2009

In this paper, a new basis for polynomial curve modeling is presented with its linear computation. This new proposed curve can be formed by the convex combination of its blending functions and related control points. Moreover, several important geometric properties for this curve are identified, for examples, a partition of unity, convex hull property and symmetry. Later the recursive algorithm, coefficient matrix representation, the derivatives and the relationships between Bézier curve and this proposed curve are defined. Finally, a new proposed rectangular and triangular basis functions are also presented with their surface definitions.

2001

We discus several alternatives to the rational Bézier model, based on using curves generated by mixing polynomial and trigonometric functions, and expressing them in bases with optimal shape preserving properties (normalized B-bases). For this purpose we develop new tools for finding Bbases in general spaces. We also revisit the C-Bézier curves presented by , which coincide with the helix spline segments developed by , and are nothing else than curves expressed in the normalized B-basis of the space P 1 = span{1, t, cos t, sin t}. Such curves provide a valuable alternative to the rational Bézier model, because they can deal with both free form curves and remarkable analytical shapes, including the circle, cycloid and helix. Finally, we explore extensions of the space P 1 , by mixing algebraic and trigonometric polynomials. In particular, we show that the spaces P 2 = span{1, t, cos t, sin t, cos 2t, sin 2t}, Q = span{1, t, t 2 , cos t, sin t} and I = span{1, t, cos t, sin t, t cos t, t sin t} are also suitable for shape preserving design, and we find their normalized B-basis. 

The Visual Computer, 2010

This paper presents a constructive method for generating a uniform cubic B-spline curve interpolating a set of data points simultaneously controlled by normal and curvature constraints. By comparison, currently published methods have addressed one or two of those constraints (point, normal or cross-curvature interpolation), but not all three constraints simultaneously with C2 continuity. Combining these constraints provides better control of the generated curve in particular for feature curves on free-form surfaces. Our approach is local and provides exact interpolation of these constraints.

Advances in Manufacturing Science and Technology

The paper presents the method of approximating curves with a single segment of the B-Spline and Bézier curves. The method for determining a single curve segment using the optimization methods in the CATIA environment is shown. The algorithms of simulated annealing and design of experiment are used for optimization. For the same purpose, a new original procedure for determining the distance between the given curves using explicit parameters in the CATIA environment was also used. This approximation of the cyclic curves results in the curve oscillation as shown in the examples. The results show that the approximation method with Bézier curve using control points as “free” points can be applied to obtain the best results of approximation.